# Properties

 Label 9680.2.a.bb Level $9680$ Weight $2$ Character orbit 9680.a Self dual yes Analytic conductor $77.295$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9680 = 2^{4} \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9680.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$77.2951891566$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 220) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 2 q^{3} + q^{5} - 4 q^{7} + q^{9}+O(q^{10})$$ q + 2 * q^3 + q^5 - 4 * q^7 + q^9 $$q + 2 q^{3} + q^{5} - 4 q^{7} + q^{9} + 4 q^{13} + 2 q^{15} - 4 q^{19} - 8 q^{21} + 6 q^{23} + q^{25} - 4 q^{27} + 6 q^{29} - 8 q^{31} - 4 q^{35} + 2 q^{37} + 8 q^{39} - 6 q^{41} + 8 q^{43} + q^{45} - 6 q^{47} + 9 q^{49} - 6 q^{53} - 8 q^{57} + 12 q^{59} - 2 q^{61} - 4 q^{63} + 4 q^{65} + 10 q^{67} + 12 q^{69} + 12 q^{71} + 16 q^{73} + 2 q^{75} + 8 q^{79} - 11 q^{81} + 12 q^{87} + 6 q^{89} - 16 q^{91} - 16 q^{93} - 4 q^{95} + 14 q^{97}+O(q^{100})$$ q + 2 * q^3 + q^5 - 4 * q^7 + q^9 + 4 * q^13 + 2 * q^15 - 4 * q^19 - 8 * q^21 + 6 * q^23 + q^25 - 4 * q^27 + 6 * q^29 - 8 * q^31 - 4 * q^35 + 2 * q^37 + 8 * q^39 - 6 * q^41 + 8 * q^43 + q^45 - 6 * q^47 + 9 * q^49 - 6 * q^53 - 8 * q^57 + 12 * q^59 - 2 * q^61 - 4 * q^63 + 4 * q^65 + 10 * q^67 + 12 * q^69 + 12 * q^71 + 16 * q^73 + 2 * q^75 + 8 * q^79 - 11 * q^81 + 12 * q^87 + 6 * q^89 - 16 * q^91 - 16 * q^93 - 4 * q^95 + 14 * q^97

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 2.00000 0 1.00000 0 −4.00000 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$
$$11$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9680.2.a.bb 1
4.b odd 2 1 2420.2.a.b 1
11.b odd 2 1 880.2.a.j 1
33.d even 2 1 7920.2.a.o 1
44.c even 2 1 220.2.a.a 1
55.d odd 2 1 4400.2.a.e 1
55.e even 4 2 4400.2.b.f 2
88.b odd 2 1 3520.2.a.d 1
88.g even 2 1 3520.2.a.bd 1
132.d odd 2 1 1980.2.a.a 1
220.g even 2 1 1100.2.a.e 1
220.i odd 4 2 1100.2.b.a 2
660.g odd 2 1 9900.2.a.bd 1
660.q even 4 2 9900.2.c.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
220.2.a.a 1 44.c even 2 1
880.2.a.j 1 11.b odd 2 1
1100.2.a.e 1 220.g even 2 1
1100.2.b.a 2 220.i odd 4 2
1980.2.a.a 1 132.d odd 2 1
2420.2.a.b 1 4.b odd 2 1
3520.2.a.d 1 88.b odd 2 1
3520.2.a.bd 1 88.g even 2 1
4400.2.a.e 1 55.d odd 2 1
4400.2.b.f 2 55.e even 4 2
7920.2.a.o 1 33.d even 2 1
9680.2.a.bb 1 1.a even 1 1 trivial
9900.2.a.bd 1 660.g odd 2 1
9900.2.c.m 2 660.q even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9680))$$:

 $$T_{3} - 2$$ T3 - 2 $$T_{7} + 4$$ T7 + 4 $$T_{13} - 4$$ T13 - 4 $$T_{17}$$ T17

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 2$$
$5$ $$T - 1$$
$7$ $$T + 4$$
$11$ $$T$$
$13$ $$T - 4$$
$17$ $$T$$
$19$ $$T + 4$$
$23$ $$T - 6$$
$29$ $$T - 6$$
$31$ $$T + 8$$
$37$ $$T - 2$$
$41$ $$T + 6$$
$43$ $$T - 8$$
$47$ $$T + 6$$
$53$ $$T + 6$$
$59$ $$T - 12$$
$61$ $$T + 2$$
$67$ $$T - 10$$
$71$ $$T - 12$$
$73$ $$T - 16$$
$79$ $$T - 8$$
$83$ $$T$$
$89$ $$T - 6$$
$97$ $$T - 14$$